Beyond Transfer Functions
While transfer functions describe input-output relationships, state-space representations capture the complete internal dynamics of a system. By expressing an nth-order differential equation as n first-order equations, state-space reveals hidden modes, internal coupling, and structural properties like controllability and observability. This formulation, pioneered by Rudolf Kalman in the 1960s, became the foundation of modern control theory.
The Phase Plane
For a two-dimensional system, the phase plane plots x₁ against x₂, with each point representing a complete system state. Trajectories flow according to the vector field defined by ẋ = Ax, spiraling inward (stable focus), diverging outward (unstable focus), or following saddle-shaped paths. The eigenvalues of A completely determine the portrait: real eigenvalues give nodes, complex eigenvalues give spirals, and pure imaginary eigenvalues give concentric orbits (centers).
Eigenvalue Classification
The trace-determinant plane classifies all possible behaviors of a 2D linear system. Below the parabola tr²=4·det lie spiral behaviors (complex eigenvalues); above lie nodes (real eigenvalues). The tr=0 line separates stable (left) from unstable (right). The det=0 line marks the boundary where one eigenvalue is zero. This simulator visualizes the eigenvalues and maps them to the corresponding phase portrait in real time.
Control Design in State Space
State feedback u = -Kx places the closed-loop eigenvalues at arbitrary locations (if the system is controllable), a vastly more powerful technique than gain tuning alone. Combined with a state observer (Luenberger or Kalman filter), this yields the separation principle: design the controller and observer independently. These ideas underpin modern autopilots, robotic controllers, and industrial process control systems.